Transformations Misconceptions

The second chapter of Putting Essential Understanding of Geometry in Grades 9-12 addresses geometric transformations.  Overall, I feel that this a unit that our students are generally successful with, but could be improved by allowing more time to practice.

The authors identified several misconceptions related to transformations, many of which I haven't noticed in my students.

1. Reflection lines must be horizontal or vertical. 
2. Reflection lines must be adjacent to a side of the pre-image. 
3. Centers of rotation must be at a vertex or a midpoint of one of the sides of a pre-image.
4. A translation may move a figure differently if it uses different points on the figure (or other points in the plane that aren't part of the figure). 
5. Isometries may not always produce images that are congruent to the preimage.
6. A transformation moves only some points in the plane. 

Things I Want to Change in My Classes:

Of the 6 misconceptions listed, my students typically struggle most with the last one. That is not to say that none of my students deal with the other issues.  I am certain some do. These are things I need to keep in mind when this unit is taught this fall.  Another thing that I've identified that needs to be addressed is more inclusion of the idea of fixed points.  In the past, fixed points have come up in discussion, but I feel as though the students didn't really understand the concept.

In an unrelated statement, I want to try to include the Match a Transformation task mentioned in this chapter this fall.  As was mentioned in my last post, I want to incorporate more opportunities for student to develop definitions and make conjectures. As part of this, I want to incorporate more use of transformations to determine congruent figures and to identify properties of figures. 

Things I Learned in this Chapter:

1. To reflect a figure about a line of the form y=x+b: Translate -b units to the line y=x, reflect over y=x, then translate b units.  (Note: This would requires that students understand the MEANING of b in the equation.)
2. To rotate about a point (a,b): translate <-a,-b>, rotate about the origin by the stated number of degrees and in the stated direction, then translate <a,b>.
3. Secondary math doesn't usually include reflections over y=ax+b.

What I found strange about this chapter is that transformations are taught in detail without using coordinates.  Transformations using coordinates after students have developed an understanding of how transformations behave.  Our unit is taught using primarily coordinates.